Connected graphs

A cellularly embedded graph ψ of one face is called a pseudo-tree. The spanning sub-graph of a cellularly embedded sub-graph G is a pseudo-tree. In chapter two we proved that µ(D(T)) = 1 where T is a tree, but in the next proof we show that

µ(D(ψ)) ≤ 3, 41

whereψ is a pseudo-tree embedded on the torus. ψ embedded on the tours can have two cycles around the handle which is not bounding a disc. The following example has two diagrams one for a pseudo-tree and another one for an embedded graph which is not a pseudo-tree.

Example

Figure 4.2 show us an example and non-example of a pseudo-tree.

Figure 4.2: (A) is a diagram for a pseudo-tree, and (B) is a diagram for a cellularly embedded graph which is not a pseudo-tree.

Theorem 22 Let ψ be a pseudo-tree on the torus. Then

µ(D(ψ)) ≤ 3.

Proof

There are several steps in this proof.

1 In this step each bridge, and each pair of non-parallel edges incident with a common vertex of degree two, are contracted. From Theorem 3 and Theorem 4, this step constructs a graphλ, with one face and the same number of components of a link diagram asψ.

2 Since λ is got by contracting edges, which is a purely local operations on ψ, then these contractions preserve faces bound discs. Thereforeλ is a cellularly embedded graph on the torus, and so it must have at least one meridian cycle and one longitude cycle. Ifλ had two or more meridian cycles, then it would have two or more faces, and this is a contradiction. So it has just one meridian cycle. Similarly, it has just one longitude cycle.

3 Let M and L denote the meridian and longitude cycles, and let p and q be the end vertices of the path M ∩ L. If M ∩ L were not connected then there would be more than one longitude or meridian. Ifλ has more than one longitude or meridian that makes new faces bound discs, which is a contradiction.

4 We show that each vertex in λ has degree 2, except for p and q which have degree 3 or 4. By step 1 M ∩ L is either one vertex or a path of one edge, because more than one edge in this path would allow for further contractions. If M ∩ L is one vertex then p = q. If the degree of p were more than four then there would be at least five edges incident with p. Four of these five edges are accounted for: two for M and two for L. The fifth edge, if it exists, cannot be a bridge because of step 1, so it must be connected with another part ofλ, which makes a second face, which is a contradiction. If M ∩ L is a path of one edge then the degrees of p and q are three.

Otherwise, we arrive at the same contradiction as in the previous case. Also for this reason, any another vertex inλ must be of degree two.

5 Suppose λ contained more than three vertices. In this case we can find pairs of edges,

each pair incident with a common vertex of degree two. But in step 1 we removed all such pairs.

6 There are just four possible graphs λ, as in figure 4.3. �

Figure 4.3: Just these four diagrams ofλ satisfy Theorem 22.

Theorem 23 is a most important theorem. It gives an upper bound for the number of components of the link diagram of a pseudo-tree embedded in the surface of genus g.

This number depends on g.

Theorem 23 Let ψ be a pseudo-tree embedded on a surface of genus g. Then µ(D(ψ)) ≤ 1 + 2g.

Proof

We are going to complete the proof by induction on the number of genus.

Suppose that g = 0. Then by Theorem 1 µ(D(ψ)) ≤ 1; ψ here is a tree.

Let ψi be a pseudo-tree embedded in a surface Si of genus g, we have µ(D(ψi))≤ 1 + 2g.

Let Si+1be a surface of genus g + 1, andψi+1be a pseudo-tree embedded in Si+1. We are going to prove:

µ(D(ψi+1))≤ 1 + 2(g + 1) = 3 + 2g.

This means that we will prove thatψi+1has at most two more components thanψi. Consider one of the handles of Si+1, and let L and M be the longitude and meridian cycles inψ_i+1for this handle, see figure 4.4.

Figure 4.4: L is the new longitude cycle and M is the new meridian cycle of Si+1surface, where eLis an edge in L, but not in M. eMis an edge in M not in L.

Choose one edge eL in L, but not in M, and then delete this edge. By Theorem 7

µ(D(ψi+1))≤ µ(D(ψi+1\ eL)) +1.

Repeat the same process with M by choosing one edge eM in M, but not in L, and we get

µ(D(ψi+1))≤ µ(D(ψi+1\ eM)) +1.

These two deletions yield a graph which is no longer ψ a pseudo-tree on S_i+1, but is a pseudo-tree on the surface of genus g obtained from Si+1 by removing the handle under consideration, as in figure 4.5. This gives

µ(D(ψi+1))≤ µ(D(ψi)) +2.

�

Figure 4.5:ψi+1\ eM\ eL.

The following theorem finds an upper bound for the number of components of a link diagram in a cellularly embedded graph by finding the relationship between the number of components of the link diagram and the number of faces in any c.e. graph.

Theorem 24 Let G be a cellularly embedded graph on a surface of genus g. Then

1 ≤ µ(D(G)) ≤ f (G) + 2g.

Proof

Suppose that G is a cellularly embedded graph in the surface of g = 0. Then G is a connected plane graph, and by Theorem 8

1 ≤ µ(D(G)) ≤ f (G) + 2g.

The theorem holds.

If this theorem is correct for any cellularly embedded graph in a surface of genus g−1, then assume G is a cellularly embedded graph in surface S of genus g. Now we need to

prove that

1 ≤ µ(D(G)) ≤ f (G) + 2g.

Letψ be a spanning pseudo-tree of G, where ψ is a cellularly embedded graph with one face. Then by Theorem 23

µ(D(ψ)) ≤ 1+2g

This means the theorem holds, becauseψ has one face. One by one, edges are added to ψ in order to get G. This insertion increases the number of faces by exactly one each time, because each new edge joins two vertices in ψ, where we have two types of new edges one when this edge is round a disc and it is clear there is a new face, another type when this edge is around a handle the new face would be bounded between two meridian or longitude cycles. Then this addition of edges gives the following sequence of graphs embedded in S.

for each i, which means that

µ(D(G)) ≤ f (G)+2g.

� We use the fact of v(G) = f (G^∗)again in a c.e. graph to prove the following corollary:

Corollary 6 Let G^∗be a cellularly embedded graph on a surface of genus g, then

µ(D(G)) ≤ v(G)+2g.

is impossible in the case of any graph containing parallel edges bounding a disc, where deleting this pair of parallel edges will produce a connected graph. This case will be proved in next theorem.

Theorem 25 Let G be a cellularly embedded graph on a surface of genus g with a pair of parallel edges a and b. If a and b bound a disc and G \ a \ b is a cellularly embedded

graph, then

µ(D(G)) ≤ f (G)+2g−2.

Proof

If a and b are parallel edges bounding a disc, and G \ a \ b is a cellularly embedded, then by Theorem 24

If G is a cellularly embedded graph on a surface of genus g, then G is calledextremal if

µ(D(G)) = f (G) + 2g.

Theorem 25 helps us to distinguish some properties of extremal graphs as in the following theorem:

Theorem 26 Let G be extremal, with a pair of parallel edges a and b bounding a disc.

Then G \ a \ b is not a cellularly embedded graph.

Proof